Papers › A correlation inequality for random points in a hypercube with some implications
A correlation inequality for random points in a hypercube with some implications
Royi Jacobovic, Or Zuk
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Let ≺ be the product order on ℝᵏ and assume that X₁,X₂,…,Xₙ (n≥3) are i.i.d. random vectors distributed uniformly in the unit hypercube [0,1]ᵏ. Let S be the (random) set of vectors in ℝᵏ that ≺-dominate all vectors in {X₃,..,Xₙ}, and let W be the set of vectors that are not ≺-dominated by any vector in {X₃,..,Xₙ}. The main result of this work is the correlation inequality P(X₂∈W|X₁∈W)≤P(X₂∈W|X₁∈S) . For every 1≤i ≤n let E_(i,n) be the event that Xᵢ is not ≺-dominated by any of the other vectors in {X₁,…,Xₙ}. The main inequality yields an elementary proof for the result that the events E_(1,n) and E_(2,n) are asymptotically independent as n→∞. Furthermore, we derive a related combinatorial formula for the variance of the sum ∑ᵢ₌₁ⁿ 1_(E_(i,n)), i.e. the number of maxima under the product order ≺, and show that certain linear functionals of partial sums of {1_(E_(i,n));1≤i≤n} are asymptotically normal as n→∞.
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